In the low temperature limit i.e. \(lT \ \&\ l^{\prime }T \ll 1\)  or equivalently \(r_H \ll {r^*}_{l^{\prime }}\ \&\ r^*_{l^{\prime }+2l}\) , at high energy, we use eqs. (REF ) and () and obtain

I(lc,l'c ,lT,l'T)4GN(5) I(l,l',lc,l'c ,lT,l'T)L2c T=1[2(lc)(lT)-1(l'c)(l'T)-1(l'c+2lc)(l'T+2lT)]-21(l'T+lT)3(l'c+lc)-23(l'c+lc)3(l'T+lT),
    1<0, 1,3>0,    
where \(\tilde{\mathcal {I}}(l\Lambda _c,l^{\prime }\Lambda _c ,lT,l^{\prime }T)\)  is the redefined \(I(l,l^{\prime },l\Lambda _c,l^{\prime }\Lambda _c ,lT,l^{\prime }T)\)  which is given by eq.(REF ) in appendix . By this redefinition the limits \(l\Lambda _c \rightarrow 0\) , \(lT\rightarrow 0\) , \(l^{\prime }\Lambda _c \rightarrow 0\)  and \(l^{\prime }T\rightarrow 0\)  become meaningful and intuitive. The first two terms, including \(\kappa _1\)  and \(\bar{\kappa }_1\) , are the known results corresponding to the pure AdS and AdS black hole, respectively and the next term, including \(\kappa _3\) , indicates the non-conformal effect. Since \(\bar{\kappa }_1\)  and \(\kappa _3\)  are positive constants the second and third terms are always negative and hence thermal fluctuations as well as non-conformal effects decrease \(\tilde{\mathcal {I}}(l\Lambda _c,l^{\prime }\Lambda _c ,lT,l^{\prime }T)\) . In this regime, the term including \(\bar{\kappa }_1\)  which is the finite temperature corrections shows that \(I(l,l^{\prime },l\Lambda _c,l^{\prime }\Lambda _c ,lT,l^{\prime }T)\)  obeys an area law. In [1]}, this behavior is generally proved. In the following, we represent the results corresponding to above limits.
